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Kibble–Zurek Mechanism in Colloidal Monolayers SEE COMMENTARY Kibble–Zurek mechanism in colloidal monolayers SEE COMMENTARY Sven Deutschländer, Patrick Dillmann, Georg Maret, and Peter Keim1 Department of Physics, University of Konstanz, 78464 Konstanz, Germany Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved March 27, 2015 (received for review January 13, 2015) – ξ The Kibble Zurek mechanism describes the evolution of topolog- called the Ginzburg temperature TG, and the length scale d of ical defect structures like domain walls, strings, and monopoles the initial (proto)domains is supposed to be equal to the corre- ξ = ξð Þ when a system is driven through a second-order phase transition. lation length at that temperature: d TG (2). The model is used on very different scales like the Higgs field in The geometry of the defect network that separates the un- the early universe or quantum fluids in condensed matter systems. correlated domains is given by the topology of the manifold of A defect structure naturally arises during cooling if separated re- degenerated states that can exist in the low symmetry phase. gions are too far apart to communicate (e.g., about their orienta- Thus, it depends strongly on the dimensionality of the system D tion or phase) due to finite signal velocity. This lack of causality and on the dimension N of the order parameter itself. Regarding results in separated domains with different (degenerated) locally the square root of Eq. 2, the expectation value of a one-com- broken symmetry. Within this picture, we investigate the nonequi- ponent order parameter (N = 1) can only take two different low librium dynamics in a condensed matter analog, a 2D ensemble of symmetry values hϕi =±ηðTÞ (e.g., the magnetization in a 2D or colloidal particles. In equilibrium, it obeys the so-called Kosterlitz– 3D Ising model): the manifold of the possible states is discon- Thouless–Halperin–Nelson–Young (KTHNY) melting scenario with nected. This topological constraint has a crucial effect if one continuous (second order-like) phase transitions. The ensemble is considers a mesh of symmetry broken domains where hϕi is exposed to a set of finite cooling rates covering roughly three chosen randomly as either +η or −η. If two neighboring (but orders of magnitude. Along this process, we analyze the defect uncorrelated) domains have the same expectation value of hϕi, and domain structure quantitatively via video microscopy and de- they can merge. In contrast, domains with an opposite expecta- termine the scaling of the corresponding length scales as a func- tion value will be separated by a domain wall in 3D (or a domain tion of the cooling rate. We indeed observe the scaling predicted line in 2D). At its center, the domain wall attains a value of – by the Kibble Zurek mechanism for the KTHNY universality class. hϕi = 0, providing a continuous crossover of the expectation value between the domains (Fig. 1A). Consider now N = 2: hϕi Kibble–Zurek mechanism | nonequilibrium dynamics | spontaneous hϕi2 = hϕ i2 + hϕ i2 = η2ð Þ can take any value on a circle, e.g., x y T symmetry breaking | KTHNY theory | colloids (all of the order parameter values that are lying on the minimum circle of the sombrero are degenerated). Because the manifold n the formalism of gauge theory with spontaneously broken of possible low symmetry states is now connected, hϕi can vary Isymmetry, Zel’dovich et al. and Kibble postulated a cosmo- smoothly along a path (Fig. 1B). In a network of symmetry logical phase transition during the cooling down of the early broken domains in two dimensions, at least three domains (in universe. This transition leads to degenerated states of vacua Fig. 1B separated by dashed lines) meet at a mutual edge. On a below a critical temperature, separated or dispersed by defect closed path around the edge, the expectation value hϕi might be structures as domain walls, strings, or monopoles (1–3). In the either constant along the path (for a global, uniform ϕ) but can course of the transition, the vacuum can be described via an also vary by a multiple of 2π (in analogy to the winding numbers N-component, scalar order parameter ϕ (known as the Higgs in liquid crystals). In the first case, the closed path can be re- field) underlying an effective potential duced to a point with hϕi ≠ 0, and no defect is built. If the path is À Á shrunk in the second case, the field eventually has to attain = ϕ2 + ϕ2 − η2 2 [1] V a b 0 , Significance η where a is temperature dependent, b is a constant, and 0 is the hϕi = modulus of at T 0. For high temperatures, V has a single Spontaneous symmetry breaking describes a variety of trans- ϕ = minimum at 0 (high symmetry) but develops a minimum formations from high- to low-temperature phases and applies to “ ” landscape of degenerated vacua below a critical temperature cosmological concepts as well as atomic systems. T. W. B. Kibble = Tc (e.g., the so-called sombrero shape for N 2). Cooling down suggested defect structures (domain walls, strings, and mono- from the high symmetry phase, the system undergoes a phase poles) to appear during the expansion and cooling of the early transition at Tc into an ordered (low symmetry) phase with non- universe. The lack of such defects within the visible horizon of hϕi < zero . For T Tc it holds the universe mainly motivated inflationary Big Bang theories. À Á W. H. Zurek pointed out that the same principles are relevant hϕi2 = η2 − 2 2 = η2ð Þ [2] 0 1 T Tc T . within the laboratory when a system obeying a second-order phase transition is cooled at finite rates into the low symmetry Caused by thermal fluctuations, one can expect that below Tc, phase. Using a colloidal system, we visualize the Kibble–Zurek hϕi takes different nonzero values in regions that are not con- mechanism on single particle level and clarify its nature in the PHYSICS nected by causality. The question now arising concerns the de- background of an established microscopic melting formalism. ξ termination of the typical length scale d of these regions and ξ their separation. For a finite cooling rate, d is limited by the Author contributions: P.K. designed research; S.D. and P.K. performed research; P.D. speed of propagating information, which is given by the finite contributed new reagents/analytic tools; S.D. analyzed data; and S.D., G.M., and P.K. speed of light defining an ultimate event horizon. Independent of wrote the paper. the nature of the limiting causality, Kibble argued that as long as The authors declare no conflict of interest. the difference in free energy ΔF (of a certain system volume) This article is a PNAS Direct Submission. between its high symmetry state hϕi = 0 and a possible finite Freely available online through the PNAS open access option. value of hϕi just below Tc is less than kBT, the volume can jump See Commentary on page 6780. 1 between both phases. The temperature at which ΔF = kBT is To whom correspondence should be addressed. Email: [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1500763112 PNAS | June 2, 2015 | vol. 112 | no. 22 | 6925–6930 Downloaded by guest on September 29, 2021 AB ^t = τð^tÞ. [3] <φ> = +η 2 2 2 <φx> +<φy> = η path The frozen-out correlation length ^ξ is then set at the temperature ^e of the corresponding freeze-out time: ^ξ = ξð^eÞ = ξð^tÞ. For a lin- ear temperature quench <φ> = 0 ‘path’ e = ðT − TcÞ=Tc = t τq, [4] =ð +μÞ with the quench time scale τ , one observes ^t = ðτ τμÞ1 1 and <φ> = 0 q 0 q <φ> = -η À Á ^ ^ ν=ð1+μÞ ξ = ξðtÞ = ξ τq τ0 . [5] Fig. 1. Emergence of defects in the Higgs field that is illustrated with red 0 vectors (shown in 2D for simplicity). (A) For N = 1 and D = 3, domain walls can D = B N = D = For the Ginzburg–Landau model (ν = 1=2, μ = 1), one finds the appear (strings for 2). ( )For 2 and 3, nontrivial topologies are ^ 1=4 strings (monopoles for D = 2). The defects are regions where the order pa- scaling ξ ∼ τq , whereas a renormalization group correction ^ 1=3 rameter ϕ retains the high symmetry phase (hϕi = 0) to moderate between (ν = 2=3) leads to ξ ∼ τq (4, 5). different degenerated orientations of the symmetry broken field. A frequently used approximation is that when the adiabatic regime ends at ^t before the transition, the correlation length cannot follow the critical behavior until τ again exceeds the time hϕi = 0 within the path, and one remains with a monopole for t when T is passed. Given a symmetric divergence of τ around = = c ^ D 2 or a string for D 3 (2, 3). A condensed matter analog Tc, this is the time t after the transition. The period in between is would be a vortex of normal fluid with quantized circulation in known as the impulse regime, in which the correlation length is superfluid helium. For N = 3 and D = 3, four domains can meet at assumed not to evolve further. A recent analytical investigation, a mutual point, and the degenerated solutions of the low tem- however, suggests that in this period, the system falls into a re- hϕi2 = hϕ i2 + hϕ i2 + hϕ i2 perature phase lie on a sphere: x y z . If the gime of critical coarse graining (6). There, the typical length field now again varies circularly on a spherical path (all field scale of correlated domains continues to grow because local arrows point radially outward), a shrinking of this sphere leads to fluctuations are still allowed, and the system is out of equilib- a monopole in three dimensions (2, 3).
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